Some places allow the current occupant to re-up until 11 AM, but the daily check-out time is 1 PM. So the campground office will not book an occupied site (that day) until after 11 AM, because the current occupant can still reserve another day (or week for that matter) up until 11 AM. Once 11 AM passes, they know the current occupant is not staying longer and they can go ahead and give it to whomever gets to the office first.
If this was the case, all the new occupant needs to do is
1. politely ask the current occupant if they intend on leaving that day or staying another day, and
2. then wait at the office at 11 AM to book the site. They could actually skip step 1, but they might then wait at the office in vain if the current occupant comes in at 10:59 to re-book.